Shell-model data from a realistic N-N interaction

i-=-l

Nuclear Physics A113 (1971) 129-140;

@ North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

SHELL-MODEL DATA FROM A REALISTIC N-N INTERACTION (II). Negative parity states in 160 and 40Ca J. M. IRVINE Department

and V. F. E. PUCKNELL

of Theoretical

Physics,

University

of Manchesier

Received 19 April 1971 Abstract: Using interaction matrix elements and a single-particle representation obtained from a Brueckner-Hartree-Fock (BHF) calculation described in an earlier work ‘) (to be referred to as I), the negative-parity states of I60 and “OCa are calculated in the particle-hole random phase approximation (RPA). In addition to the energies of the negative-parity states, the results of calculations of the ground state electric multipole transitions are also reported.

1, ~~oduction We have used the Cornell local density-dependent approximation to the nuclear reaction matrix ‘,‘) based upon the Reid soft-core interaction “) to carry out BHF calculations on the nuclei 4He, ‘He, 160, “0, 40Ca and 41Ca. The results have been reported in I. The two-body interaction matrix elements and the single-particle representations which we have obtained are intended to be used as input data for nuclear shell-model calc~ations. We have chosen as a first test of the data to carry out particlehole RPA calculations for the negative-parity states of 160 and 40Ca. We construct particle-hole creation operators A&, JT of the form

where&mp, .A,-mrJ JM) are Condon-Shortley

‘) Clebsch-Gordan coefficients, and conjugate operators _?I,h,JThaving the same transfo~a~on properties under rotations in three-space and charge-space

In terms of these two sets of operators we define a set of RPA quasiparticle creation operators “) e JT = ,cb ~X,,,rA;,r .

-(-f)=+=YplrJT~~b.r*1.

13)

For I60 the hole space consists of the Opt. and Op+ states while the particle space includes the Od+, Is* and Od, states. For 40Ca the hole space consists of the Od,, Is+ and Od, states while the particle space includes the Of*, lp+, Of+ and Ip+ states. 129

J. M. IRVINE AND V. F. E. PUCKNELL

130

Our BHF ground state contains many correlations represented by multiparticlemultihole configurations. To see this explicitly we can use the Bethe-Goldstone “) equation to write the two-body correlation functionf(x,, x2) in the form ‘)

(4 where u is the bare N-N interaction, Q is the usual Pauli projection operator 1 -I@,,) <@el with @,, the uncorrelated many-body ground state and e is the usual BHF energy denominator ‘). Thus in the Jastrow s)-Moszkowski ‘) approximation our BHF ground state may be written Yc(x, . . . i>j

(5) Thus

(6) as it would have done had Y. been the Hartree-Fock

state ‘.

2. Energy eigenstates In this work we have not explicitly calculated the amount of ground state configuration mixing - such a study is under way and will be reported on shortly - instead we have made the usual RPA assumptions and linearised the equations of motion lo), i.e. we have assumed that the excited states 1JT) of our nuclei are constructed by coupling non-interacting quasiparticles to our correlated ground state IJV

Q:#o>

=

(7)

and

QrA’f’o) = 0, Linearising the equations must satisfy the equations:
&,,]I

of motion

requires

‘f’o> = (E,,-&,,,n+ + c

D;;;;’

all

J, T.

(8)

that our coefficients

X,,,,

and Y,,,,,

C CJ,~~~‘<(p’h)J’T’lgl(ph’)J’T’)Xp,h,,T

<(hh’)J’T’lsl(pp’)J’T’)

Yp’h’J1. =

En

XphJTr

(9)

and
&&JT)

=

(Ep-%)Yph,T

+

1

C~~~~‘<(P’l~)J’~‘lgl(ph’)J’T’)Yp,h,,T

+ 2 D;;;;~<(hh’)J’T’(gl(pp’)J’T’)X~,~,~r

= -E,,

Yp,,,r,

(10)

7 Of course, a Hartree-Fock calculation would be impossible for an interaction with a strong shortrange repulsion like the Reid potential.

SHELL-MODEL

131

DATA

where

CJTJ'T' -(25’+1)(2T’+l) php’h’ =

(;;

;i

;j(:

“t

;) ,

(11)

and o;;;$,

= (-1>‘“-“-J’-~‘[(1+6pp,)(1+-6h,.)]* x (25’+ 1)(2T’+ 1) 1;:

;;

;,](;

;

;,),

(12)

where (ii /;‘;:I are 6j symbols as defined by Edmonds ‘I). The eqs. (9) and (10) may be written in the vector form ‘.JT gJT = EJT~;;. (13) php’h’ bp’h If there are N particle-hole states then %?is a 2Nx 2N matrix and 2;;: is the 2Ncomponent vector (ph) = 1. ..N (14) (N + 1) 5 (ph) $ 2N, V is a non-symmetric

matrix of the form V=

(_:

(15)

-J)T

where ~2 and &Yare symmetric N x N matrices. Since it is computationally diagonalize symmetric matrices we proceed as follows “): (i) Construct matrices V’ which are N x N and symmetric %* = .%?+a.

easier to

(16)

(ii) Construct matrices A? and $’ which are N x N and lower and upper triangular respectively vi = L?.P. (17) (iii) Construct S’P which is N x N and symmetric SF = _!W-3,

(18)

then &’ has N component eigenvectors Y and the corresponding eigenvalues A which are i = E2. (19 (iv) Construct the matrix A which is N x N and upper triangular AP=l,

(20)

then the eigenvectors of ‘37are Xi = 5(1El)~

~j (21)

132

J. M. IRVINE AND V. F. E. PUCKNELL

and

(22) Strictly speaking, the above construction is only valid if V has only positive roots. In both I60 and 40Ca the lowest T = 0, l- state resulting from a RPA calculation is totally spurious and hence in those cases the 2N x 2N matrices V,, were diagonalised directly to give the energy eigenstates. It should be noted that the eigenstates are normalised such that Cph(X& - Y&) = 1. 3. Ground state transition rates For the purposes of calculating electric transition rates we make an isotopic spin component decomposition of eq. (3) and rewrite it in the form

where JG;!h

=,~~O’Pmpjh-MhlJMJ)(-l)jh-mhaptah,

Jf?-JMJ PhVh

=

;-

=

(Jt,

XJT Tp%

and

(24)

1yw;,;;:,

(25)

TMT)(-

+-z~I

l)‘-‘“Xi,’

(26)

JT Yrpfh = (~t,t-~~lT-Mr>(-l)~-~~~~-~~YpJhT.

(27)

We define the 8N component vectors

and

Q=

(QL,I~TM~~

(-l)J-MJ’T-MTQ~-~J~-~T),

(29

in terms of which eq. (23) and its conjugate may be written Q=X*&,

(30)

where X is the 8N x 8N matrix JMJTMT x

=

_(_l)J-M~;J-M~T-M~,

_(_I)J+T~JMJTMT (_~)T-MT~J-MJT--MT

.

The inverse relationship &=xX-‘.Q

for a canonical transformation

is obtained with

(_l)T-MT~J-&T-MT,

x-1

=

(_

(32)

~)J-MJ~J-MJT-MT,

(_l)J+TyJM.rTM~ (33)

SHELL-MODEL

The ground

state electric multipole

where the transition

operator

133

DATA

transition

rate of order L may be written

is

(35) where e, is the effective charge associated sitions we are interested in

with the state Ii). For the ground

u; Uh = c ( j, nrP j, - rn,JJMJ)(

- l)j~?!zz~P~~+

JMJ

state tran-

(36)

and ala,,

= c (j,m, JMJ

Using the inverse relations

jh-mhl~-MJ)(-l)ih-mh+J-MJ~~~~~.

(32) we can write the reduced

electric multipole

(37) transition

rate

x (-

l)ih-mh+T-MTxJ-MJT-MTQ~MJTMT+(hleh

X JG
rLYL,,lp)

jh-mbl~-~J)(-l)i”-mh-M.‘-MTyJ-MJT-~~Q~~~~~~}lY/~)IZ,

(38)

where we have made use of eq. (8) Finally, eq. (38) reduces to

~ 1g WL) = (2L;l)

( j,llrLYLll jh)ep(+2,~-ThlTO>(

-

l)‘-rh(X$T+YpJhT)12* (39)

4. Results 4.1.

OXYGEN

16

In fig. 1 we reproduce a comparison between our calculated negative-parity energy levels and those observed experimentally in I60 . Using the calculated single-particle spectrum and the matrix elements quoted in I we obtain column (a). It is clear that the whole T = 0 spectrum is shifted up by approximately 8 MeV compared with experiment. The T = 1 spectrum is compressed compared with experiment with the lowest state some 6 MeV higher than observed. Using the experimental single-particle spectrum given in I we obtain column (b) in which the discrepancy between the theory and experiment is greatly reduced. The T = 0 spectrum is still higher than experiment by 2-3 MeV and there is a serious discrepancy in the low-lying T = 0,l- spectrum. There is always considerably greater uncertainty in interpreting the single-particle intershell spacing than there is in interpreting the single-particle spacing within a major shell 13,14). Therefore we have repeated the calculations reported above but

3-

,-

2-

232-

41-

o-

(a)

-1 73

-I

-l

-I

-I

T=O J

0

2

T=l J

3-

l-

3-

I-

30-

2s: I-

2-o

3-

S-

I2-

(b)

-? ---J

-l

I-

23o-

5-I P32-

4-

O-

-I

-I

-I

(cl

-1

T=O J

0

T=l J

Fig. 2. Spectra of negative-parity states in 160; (a) using the calculated single-particle spectrum of I with the particle-hole gap reduced by 3.7 MeV, (b) using the experimental single-particle spectrum quoted in I with the particle-hole gap reduced by I.8 MeV and (c) from experiment.

O-

5.

IO.

$15,

20.

25.

30-

3-

I-

o-

2-

2-f

423-

I-1

(a)

-b

-I

-I

-I

T=O

J

T=l

I-

3-

I-

20-

2-

23-

-

-3

-l

-I

-I

T=O J

(b)

0

:

T=l J

3-

5-

3-

I-

2-

I-

2x: I3o-

-1

-I

-I

-l-=3

2-3

J

(cl

0

T=l J

Fig. 1. Spectra of negative-parity states in 160; (a) using the calculated single-particle spectrum of I, (b) using the experimental single-particle spectrum quoted in I and (c) from experiment.

J

SHELL-MODEL

135

DATA

treating the p+-d, energy spacing as a parameter fitted to give a best fit to experiment. When the calculated spectrum with the particle-hole gap reduced by 3.7 MeV is used we obtain column (a) in fig. 2. The base of the T = 0 and T = 1 spectra now occur at their observed energies but the T = 0 spectrum is slightly extended compared with ,

3o

T=O J

T=I J

T=O J

T= I

T=l J

J

-I

-1

25.

T=l J

-1

-I -1 20.

-I I- -I 38E5E: 2OI-

2

-1

-1 b

Ip-----32-

1

5P

-I

3-

IS-

I 2oI-

3-

----A -2

3

2=7 :--

8

3P o-

IO-

,2-

3-

,3-

s-

O-

(al

b)

(c)

Fig. 3. Spectra of negative-parity states in 160; (a) using the experimental single-particle spectrum with the interaction increased by 7 %, (b) as in (a) but including a low-lying K = l- rotational band and (c) from experiment.

experiment and the lowest J = I and J = 2 states are not explained. The T = 1 spectrum is still compressed compared with experiment and this leads to a serious discrepancy between the observed and the calculated positions of the giant dipoie states. If the experimental single-particle spectrum is employed with the particle-hole gap reduced by 1.8 MeV we obtain column (b) in fig. 2. The lowest T = 1 state lies about 1 MeV below their observed position but a reasonable description of the giant dipole states is obtained. The T = 0 spectrum is extended compared with experiment and apart from the lowest 3- state the comparison with experiment is not very good. We know from I that our matrix elements do not produce sufficient ground state binding energy and that indeed an increase of approximately 20 % is required in the Cornell interaction if it is to bind nuclear matter at the energy predicted by the semiempirical mass formula. If we increase the strength of our interaction not only does

136

J. M. IRVINE

AND V. F. E. PUCKNELL

the ground state binding energy improve but the calculated single-particle spectrum also becomes similar to the experimental single-particle spectrum. A study of these effects coupled to an improved treatment of the starting energy problem is in progress. In fig. 3 we reproduce the results of a calculation in which the experimental spectrum of I has been used along with our calculated matrix elements increased by 7 %. The results are in fair agreement with experiment. The principal discrepancy is still in the low-lying T = 0,l- spectrum although the second T = 0,3- state is also not particularly well represented. It is clear that in order to explain the low-lying T = 0 spectrum it is necessary to introduce states of a completely different structure from those already employed. The three-particle-three-hole states lie in this region but the inclusion of these states alone is not sufficient to produce a satisfactory explanation of the observed spectrum “). There has been the suggestion that the l- state at 9.59 MeV may be the base of a K = l- rotational band. In column (b) of fig. 3 we introTABLE 1

Some ground state transition rates in Weisskopf units for the dominant decays in I60 Calculated excitation energy J

T

WW

3 3 1 1 1 1 1

0 1 1 1 1 1 1

6.15 13.59 13.68 18.04 20.89 23.55 27.10

Experimental

reduced transition rate (W.U.)

excitation energy

6.55 0.94 0.14 0.002 0.37 8.36 0.58

6.14 13.26 13.1 17.3

(MeV)

reduced transition rate (W.U.) 13.8 0.21

22.3 w 25

The calculated rates are for an effective charge of #e on protons and -+e trum of fig. 3(a).

on neutrons and the spec-

duce such a rotational band and assume that it mixes with our RPA states to reproduce the lowest observed 1- state. The resulting spectrum is in very good agreement with experiment, the worst discrepancy now being in the lowest 2- state which is some 2 MeV higher than observed. The states most affected by the mixing in of a rotational band show no large ground state transition strengths and this feature will be preserved because of the necessarily more complicated structure of the rotational states. In table 1 we present our calculated ground state electric transition rates for an effective proton charge + se and neutron charge - $e. Strictly speaking, such an effective charge is only justified for the dipole transitions 16) and a L-dependent effective charge should be employed.

h fig. 4 we compare QUCcalculated s~~~tr~rn with ~x~rirn~~t. If we use the talculated sjngle-particle $~~~urn of I, then both the 7” = 0 and I” = 1 states lie much too high in energy. Using the e ~irnent~~s~~~~-p~rticle s~ctr~rn there is a distinct improvement in the T = 1 ~~~~~rn aith~~g~ the fewest 2- state is still G 1 Me%’

higher than observed. The whole T = 0 s~~trum, however, is shifted up by z 3 MeV ~~rn~~~~d with ex~rimeut. the p~ti~le-~o~~ gasp (Le. d&-f%s~a~jng~ as a parame~r we obtain fig. 5. cuiated si~g~e-~~~~~e spectrum with. the gap rcd~~d by 6.5 MeY we obtain the base of the T = 0 s~~t~urn at its observed position but this i~~odu~s an ~ao~~~ed low-lying OmW state and leaves the Iowest 4- state 2 MeV higher than observed. The whole T = 1 s~~t~~rn is Iower than observed by some 2 MeV and the calculation yields an unobserved low-lying O’%state while the lowest 5- and 2- states are trans~sed. Using the ex~rimen~l Si~gIe-~arti~l~ spectrum with the ~r~~~e-ho~e gap reducrcd by I MeV the low-hying T = 0,3 - and 5 - states are well rep~odu~d but the rest of the spectrum is too extended when ~rn~a~~d with experiment. The T = f spectrum lies approximately 2 MeV lower than observed, In fig. 6 we show the wufts of a ~a~~~~~~~~~rn~~oying the: ex

-8

-2

-3

(al

3

T=l

-

-I

3-

5-

:-

tb)

3_ ssiEiiF:

3-5 = ?O

I2 mp

4-

2’ -

J

T=O

;2

J

T=l

3-

J

T=O

(cl

J

T=l

Fig. 5. Spectra of negative-parity states in “Ca; (a) using the calculated single-particle spectrum of I with the particle-hole gap reduced by 6.5 McV, (b) using the experimental single-particle spectrum with the partitle-hole gap reduced by 1 MeV and (c) from experiment.

3-

J-

3

,d--_”

3-

,2 -23

4=.,

2I-

J

T=O

FO

-I

-I

J

T=l

3-

5-

4m I332I4-

fb)

-5 -2 -3

T=O

J

4

J

T-l

Fig. 6. Spectra of negative-parity states in *OCa; (a) using the experimental single particle spectrum with the interaction increased by 15 %, (b) from experiment.

3-

s-

J

SHELL-MODEL

Some ground state electric transition

TABLE 2 rates in Weisskopf

J

T

WW

3 5 3 5 3 3 1 3 3 1

0 0 0 1 1 1 1 0 1 1

3.65 4.63 7.12 8.67 9.76 11.24 12.13 12.16 18.22 18.57

units for the dominant

decays in 40Ca

Experimental

Calculated excitation energy

139

DATA

reduced transition rate (W.U.) 17.94 14.63 2.34 7.37 1.31 1.96 1.01 1.48 18.09 16.01

excitation energy (MeV)

reduced transition rate (W.U.)

3.73 4.49 6.28 8.54 7.69

15.0 12.3 1.6 6.9

6.58

1.1

The calculated rates are for an effective charge of Qe on protons and -+e on neutrons and for the spectrum in fig. 6(a).

particle spectrum with the interaction increased by 15 %. There is good agreement for the low-lying T = 0 and T = 1 spectra although the lowest T = 0,4-state is 13 MeV too high and there is an unobserved O- state predicted at z 9 MeV in the T = 1 spectrum. The ground state transition strengths are not extremely sensitive to the single-parttitle spectrum or the strength of the interaction. The calculated transition strengths are compared with the known experimental values in table 2 where it will be observed that the dominant decays are again well reproduced. One of us (V.F.P.) acknowledges the receipt of an SRC Studentship during the period of this work. Computing facilities were provided at the ATLAS Computing Laboratory of the University of Manchester.

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

J. M. Irvine and V. F. E. Pucknell, Nucl. Phys. A159 (1970) 513 J. Negele, Phys. Rev. Cl (1970) 1260 R. V. Reid, Ann. of Phys. 50 (1968) 411 E. U. Condon and G. H. Shortley, Theory of atomic spectra (Cambridge Univ. Press, London, 1925) A. E. L. Dieperink et al., Nucl. Phys. All6 (1968) 556 H. A. Bethe and J. Goldstone, Proc. Roy. Sot. A238 (1957) 551 B. Day, Rev. Mod. Phys. 39 (1967) 719 R. Jastrow, Phys. Rev. 98 (1955) 1479 S. A. Moszkowski, Phys. Rev. 140 (1965) B283

140

J. M. IRVINE AND V. F. E. PUCKNELL

10) A. M. Lane, Nuclear theory (Benjamin, New York, 1964) 11) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press, Princeton, NJ, 1957) 12) B. Chi, Nucl. Phys. Al46 (1970) 449 13) H. G. Benson and J. M. Irvine, Proc. Phys. Sot. 89 (1966) 249 14) W. Gerace, Nucl. Phys. A93 (1967) 110 15) P. J. Ellis, private communication 16) D. J. Thouless, Quantum mechanics of many-body systems (Academic Press, New York, 1961)